Cosmic string loop production functions

Author(s)

Auclair, Pierre, Ringeval, Christophe, Sakellariadou, Mairi, Steer, Daniele

Abstract

Numerical simulations of Nambu-Goto cosmic strings in an expanding universe show that the loop distribution relaxes to an universal configuration, the so-called scaling regime, which is of power law shape on large scales. Precise estimations of the power law exponent are, however, still matter of debate while numerical simulations do not incorporate all the radiation and backreaction effects expected to affect the network dynamics at small scales. By using a Boltzmann approach, we show that the steepness of the loop production function with respect to loops size is associated with drastic changes in the cosmological loop distribution. For a scale factor varying as a(t)  tν, we find that sub-critical loop production functions, having a Polchinski-Rocha exponent χ < (3 ν − 1)/2, yield scaling loop distributions which are mostly insensitive to infra-red (IR) and ultra-violet (UV) assumptions about the cosmic string network. For those, cosmological predictions are expected to be relatively robust, in accordance with previous results. On the contrary, critical and super-critical loop production functions, having χ ≥ (3ν−1)/2, are shown to be IR-physics dependent and this generically prevents the loop distribution to relax towards scaling. In the latter situation, we discuss the additional regularisations needed for convergence and show that, although a scaling regime can still be reached, the shape of the cosmological loop distribution is modified compared to the naive expectation. Finally, we discuss the implications of our findings.

Figures

Difference between loop distributions in the radiation era generated by a Dirac distribution LPF (green lower curve) and a super-critical, IR-regularised, Polchinski-Rocha one (purple top curve). Given a super-critical power-law loop production function, one can reproduce the large scale behavior of the loop distribution with a Dirac distribution for the loop production function (see section \ref{sec:deltaf}). Doing so, one loses the small-scale behavior of the loop distribution. For illustration purposes, we have chosen $GU=10^{-7}$, $c\simeq0.25$ and $\gammai=0.1$ for the super-critical LPF and $c \simeq 5.7$ for the Dirac distribution.

Difference between loop distributions in the radiation era generated by a Dirac distribution LPF (green lower curve) and a super-critical, IR-regularised, Polchinski-Rocha one (purple top curve). Given a super-critical power-law loop production function, one can reproduce the large scale behavior of the loop distribution with a Dirac distribution for the loop production function (see section \ref{sec:deltaf}). Doing so, one loses the small-scale behavior of the loop distribution. For illustration purposes, we have chosen $GU=10^{-7}$, $c\simeq0.25$ and $\gammai=0.1$ for the super-critical LPF and $c \simeq 5.7$ for the Dirac distribution.


Sketch of possible loop production function shapes under the gravitational backreaction length scale $\gammac\equiv\ellc/t$ (logarithmic units), namely $\calP(\gamma \le \gammac,t) = \cc\, \gamma^{2\chic-3}$ where the constant $\cc$ is chosen such that $\calP$ is continuous at $\gamma=\gammac$. According to Ref.~\cite{Polchinski:2007rg}, minimal gravitational backreaction effects correspond to $\chic=1$ and we take this value as a motivated lower bound. The larger the value of $\chic$, the sharper the cut is.

Sketch of possible loop production function shapes under the gravitational backreaction length scale $\gammac\equiv\ellc/t$ (logarithmic units), namely $\calP(\gamma \le \gammac,t) = \cc\, \gamma^{2\chic-3}$ where the constant $\cc$ is chosen such that $\calP$ is continuous at $\gamma=\gammac$. According to Ref.~\cite{Polchinski:2007rg}, minimal gravitational backreaction effects correspond to $\chic=1$ and we take this value as a motivated lower bound. The larger the value of $\chic$, the sharper the cut is.


Schematic representation of the different domains of $\gamma$ for $t<\tc$ and for $t > \tc$. The black regions are causally disconnected from the cutoff at $\gammai$ such that the solutions are exactly the same as the non-regularised ones. On the contrary, this is not the case in the red dotted regions and one has to use the modified expression for $t^4\calF(\gamma \ge \gammap,t)$ (see text).

Schematic representation of the different domains of $\gamma$ for $t<\tc$ and for $t > \tc$. The black regions are causally disconnected from the cutoff at $\gammai$ such that the solutions are exactly the same as the non-regularised ones. On the contrary, this is not the case in the red dotted regions and one has to use the modified expression for $t^4\calF(\gamma \ge \gammap,t)$ (see text).


Loop number density distribution at various redshifts for a critical loop production function having $\chirad=\chicrit=0.25$. The network is assumed to be formed at $\zini=10^{18}$ and $c=0.03$. At redshift $z=10^{17}$, the loop distribution is not yet fully relaxed from the initial conditions. For later redshifts, $z<10^{15}$, the non-scaling logarithmic divergence becomes clearly visible for all loops larger than the gravitational wave emission scale, $\gamma \ge \gammad$. The smaller ones, having $\gamma < \gammad$, remain in a transient scaling for most of the cosmological evolution, until the non-scaling behaviour takes over (see text).

Loop number density distribution at various redshifts for a critical loop production function having $\chirad=\chicrit=0.25$. The network is assumed to be formed at $\zini=10^{18}$ and $c=0.03$. At redshift $z=10^{17}$, the loop distribution is not yet fully relaxed from the initial conditions. For later redshifts, $z<10^{15}$, the non-scaling logarithmic divergence becomes clearly visible for all loops larger than the gravitational wave emission scale, $\gamma \ge \gammad$. The smaller ones, having $\gamma < \gammad$, remain in a transient scaling for most of the cosmological evolution, until the non-scaling behaviour takes over (see text).


Schematic representation of the different domains of $\gamma$ for $t<\tc$ and for $t > \tc$. The black regions are causally disconnected from the cutoff at $\gammai$ such that the solutions are exactly the same as the non-regularised ones. On the contrary, this is not the case in the red dotted regions and one has to use the modified expression for $t^4\calF(\gamma \ge \gammap,t)$ (see text).

Schematic representation of the different domains of $\gamma$ for $t<\tc$ and for $t > \tc$. The black regions are causally disconnected from the cutoff at $\gammai$ such that the solutions are exactly the same as the non-regularised ones. On the contrary, this is not the case in the red dotted regions and one has to use the modified expression for $t^4\calF(\gamma \ge \gammap,t)$ (see text).


Scaling loop distribution in the radiation and matter era for $\mu>0$, which corresponds to $\chi < (3\nu -1)/2$. The values for $\gammad$ and $\gammac$ are illustrative only.

Scaling loop distribution in the radiation and matter era for $\mu>0$, which corresponds to $\chi < (3\nu -1)/2$. The values for $\gammad$ and $\gammac$ are illustrative only.


Growing loop distribution generated by a super-critical loop production function having $\chi=0.45$ during the radiation era. The string tension has been set to $\GU=10^{-7}$ and the initial conditions are arbitrarily set at $\zini=10^{18}$ with $\calNini(\ell)=0$ and $c=0.14$. At redshift $z=10^{7}$, the change of shape associated with gravitational wave backreaction becomes washed out by the number loops which diverges with time.

Growing loop distribution generated by a super-critical loop production function having $\chi=0.45$ during the radiation era. The string tension has been set to $\GU=10^{-7}$ and the initial conditions are arbitrarily set at $\zini=10^{18}$ with $\calNini(\ell)=0$ and $c=0.14$. At redshift $z=10^{7}$, the change of shape associated with gravitational wave backreaction becomes washed out by the number loops which diverges with time.


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